A square wave approximated with just 3 Fourier terms. The epicycles trace the partial sum in real time.
The Core Idea
Joseph Fourier's radical claim — later proved rigorously — was that any periodic function, no matter how jagged or discontinuous, can be decomposed into a sum of sines and cosines. A square wave is not "square" at all; it's an infinite chorus of harmonics, each oscillating at an integer multiple of the fundamental frequency, their amplitudes carefully chosen so the sharp corners emerge from constructive interference.
The Mathematics
The Fourier series of a periodic function f(t) with period T is:
f(t) = a₀/2 + Σ [aₙ cos(2πnt/T) + bₙ sin(2πnt/T)]
where the coefficients are:
aₙ = (2/T) ∫₀ᵀ f(t) cos(2πnt/T) dt
bₙ = (2/T) ∫₀ᵀ f(t) sin(2πnt/T) dt
For a square wave (odd symmetry, so all aₙ = 0):
f(t) = (4/π) [sin(ωt) + sin(3ωt)/3 + sin(5ωt)/5 + ...]
Only odd harmonics appear, each with amplitude 1/n. This 1/n decay is slow — that's why you need many terms to build a convincing square edge.
The Gibbs phenomenon
Notice the persistent overshoot at the corners? That's the Gibbs phenomenon — a mathematical inevitability. No matter how many terms you add, the partial sum overshoots by approximately 9% of the jump height at each discontinuity. The overshoot doesn't disappear; it just gets narrower. This is not a failure of Fourier analysis — it's a deep result about the tension between pointwise and uniform convergence.
Sawtooth wave with 5 terms. All harmonics contribute, making it harder to approximate than the square wave.
Different shapes, different spectra
Switch the shape control to see how the harmonic recipe changes:
- Square wave: only odd harmonics,
1/namplitude — the classic. - Sawtooth: all harmonics,
1/namplitude — richer in high frequencies. - Triangle: only odd harmonics,
1/n²amplitude — the1/n²decay makes it smoother, needing fewer terms.
Why epicycles?
The visualisation uses rotating circles (epicycles) because each Fourier term Aₙ sin(nωt + φₙ) is exactly a vector of length Aₙ rotating at angular velocity nω. Stack them tip-to-tail and the final point traces the partial sum. Ptolemy used the same geometry for planetary orbits — Fourier just gave it rigorous mathematical footing.
Further reading
- Fourier series (Wikipedia)
- 3Blue1Brown, But what is a Fourier series? — the definitive visual explanation.
- Stein & Shakarchi, Fourier Analysis: An Introduction, Princeton Lectures in Analysis.